It's funny, when I wear a hat here in Japan and meet someone, I don't know wether to tip my hat or bow. I usually end up doing both in an awkward hat-taking-off, hide-behind-my-back and bow ceremony. I don't think Japanese natives have that issue because they weren't taught to take off their hat. They just leave it on and bow.

In the west you would probably bow with your hat held over your chest, left arm behind you (like at the end of a theatrical performance), but in Japan your arms should be at your sides or slightly behind you.